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arXiv:1405.3967v1 [hep-ph] 15 May 2014 Towards an effective action for relativistic dissipative hydrodynamics

Pavel Kovtun

1, Guy D. Moore2, and Paul Romatschke3

1 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada

2Department of Physics, McGill University, 3600 rue University, Montr´eal, QC H3A 2T8, Canada

3Department of Physics, 390 UCB, University of Colorado, Boulder, CO 80309-0390, USA

(May 2014)

Abstract

We propose an effective action for first order relativistic dissipativehydrodynamics that can be used to evaluaten-point symmetrized correlation functions, taking into account thermal fluctuations of the hydrodynamic variables.

1 Introduction

The study of fluid dynamics is a centuries-old discipline that still seems nowhere near clo- sure. The "classical" relativistic fluid dynamics can be derived by requiring conservation of the energy-momentum tensor and global symmetry currents [

1]. In modern understanding,

these conservation equations should be constructed order by order in the derivative expansion of the hydrodynamic variables, similar to the derivative expansion in effective field theory. Truncating the derivative expansion at first order, second order and higher order one obtains ideal fluid dynamics (Euler equations), viscous fluid dynamics (Navier-Stokes equations), and higher-order dissipative hydrodynamics [

2,3]. The classical hydrodynamic correlation func-

tions can then be obtained by varying with respect to external sources, see e.g. [ 4]. In non-relativistic fluids, it is well known that there are correlation functions of the hy- drodynamic variables which can not be reproduced by such classical hydrodynamic equations, even when the frequency and momentum are arbitrarily small [

5]. The reason is that while the

classical equations describe flows generated by external sources, they neglect hydrodynamic excitations generated by thermal fluctuations within the fluid. These effects may be taken into account by supplementing the classical hydrodynamic equations with stochastic noise terms whose correlation functions are taken to be Gaussian white noise [

5]. It is natural to expect

that a similar stochastic modification is required for relativistic fluids in order to correctly reproduce physical observables. In linear non-relativistic hydrodynamics, such noise terms were introduced long ago by

Landau and Lifshitz [

6]. When considering the full non-linear theory of stochastic hydrody-

namics, one finds that the interactions lead to changes in thebasic parameters of the classical theory, such as the shear viscosity coefficient [

7]. More generally, correlation functions eval-

uated in stochastic hydrodynamics will differ from their classical counterparts by fluctuation corrections involving loops of the hydrodynamic modes. Stochastic equations for hydrody- namic variables can be readily converted to a functional integral form [

8], providing one with

1 an effective field theory. The purpose of this note is to write down an effective action for dissipative relativistic fluids. We emphasize that our interest is not in an action that will give rise to the classical hydro- dynamic equations upon using a variational procedure. Rather, we are interested in an action which can be used in a standard way in the functional integralto evaluate hydrodynamic correlation functions. While it is straightforward to derive such an effective action for the linearized viscous relativistic hydrodynamics [

4], the full non-linear hydrodynamics and the

derivative expansion require more work. The fields in the effective theory include the hydro- dynamic variables (fluid velocity, temperature etc), and wewill refer to this effective theory as "statistical hydrodynamics", to distinguish it from classical hydrodynamics which ignores fluc- tuations. The 1PI effective action of statistical hydrodynamics should give rise to the classical hydrodynamic equations at tree level, but will contain corrections to classical hydrodynamics once the loops are taken into account. The loops here are not the quantum loops (as one is not quantizing the classical hydrodynamics), but rather reflect statistical fluctuations of the hydrodynamic variables. We pause to comment on previous work addressing related questions. A variational for- mulation of classical ideal relativistic hydrodynamics (neglecting the derivative expansion, fluctuations, and dissipation) is an old subject discussed by many authors in various forms, see e.g. [

9,10,11]. As mentioned above, we don"t expect such classical constructions to be

helpful for statistical hydrodynamics. Refs. [

12,13] studied effective actions for relativistic

fluids, taking into account the derivative expansion, however the resulting effective action only captured non-dissipative information. Similarly, Refs. [

14,15] derived generating func-

tionals of relativistic fluids coupled to external sources in equilibrium. Again, this allowed a systematic construction to any order in the derivative expansion, but only captured static non- dissipative physics. For variational approaches aiming toincorporate dissipation in classical hydrodynamics, see e.g. [

16,17,18].

Recently, there have also been efforts to understand dissipation in relativistic statistical hydrodynamics (with fluctuation corrections), partly motivated by the experimental study of the quark-gluon plasma in heavy-ion collisions. Refs. [

19,20,21,4] looked at statistical

one-loop corrections to the shear viscosity, but lacked a systematic field-theoretic framework. See [

22,23,24,25] for other recent work on relativistic fluctuating hydrodynamics, including

the Israel-Stewart formulation. It is worth pointing out that the fluctuation corrections render the derivative expansion in purely classical relativistichydrodynamics ill-defined [

21]. Clearly,

one needs a unified calculational framework that takes into account the full non-linearity of relativistic hydrodynamics, the derivative expansion, and fluctuations of the hydrodynamic variables. The present paper is a step in this direction.

2 Noisy hydrodynamics

2.1 Setup

Classical relativistic hydrodynamics [

1] is a set of partial differential equations for the hy-

drodynamic fieldsuμ(x),T(x), and (for fluids with a globalU(1) charge)μ(x). Collectively denoting these hydrodynamic fields asφ, we will write the classical hydrodynamic equations in the formEa(φ) = 0, whereEμ=∂νTνμ cl,Ed+1=u2+1,Ed+2=∂μJμ cl, anddis the number of spatial dimensions. HereTμν clandJμ clare the (symmetric) energy-momentum tensor and 2 theU(1) global symmetry current, given in terms ofφ. The fluid velocity is normalized1as u

2=-1,Tis the temperature, andμis the chemical potential. The constitutive relations

expressingTμν clandJμ clin terms ofφare normally written in a given "frame" (a particular

out-of equilibrium definition ofφ), to a given order in the derivatives ofφ. The starting point

for the stochastic hydrodynamics is the modification of the classical hydrodynamic equations by "noise" terms which are interpreted as microscopic stresses and currents [

6], so that the

hydrodynamic equations take the form∂μTμν= 0, and∂μJμ= 0, whereTμν=Tμν

cl+τμν, and J

μ=Jμ

cl+rμ. The microscopic contributionsτμν(φ,ξ) andrμ(φ,ξ) are functionals of both the

hydrodynamic fieldsφand the noise fields collectively denoted asξ, so that the hydrodynamic equations become stochastic equations E a(φ) +fa(φ,ξ) = 0,(2.1)

wherefμ=∂ντνμandfd+2=∂μrμ. The form of the forcefaand the dynamics of the noise

fields need to be determined by the problem at hand. In particular, they must be such that the fluctuation-dissipation theorem is satisfied in equilibrium.

One can convert Eq. (

2.1) to a functional integral form. Let us denote the solution to

Eq. (

2.1) asφξ. Upon solving Eq. (2.1), the energy-momentum tensor and the current will

become functionals of the noise,Tμν[φξ,ξ],Jμ[φξ,ξ]. For a general functionO(φξ) we have

O(φξ) =?

Dφ δ(Ea(φ) +fa(φ,ξ))J(φ,ξ)O(φ), where the Jacobian isJ= detδ(Ea+fa) δφb. If the dynamics ofξis independent ofφ, so that the noise average is performed with someφ-independent actionSn[ξ], the correlation functions can be written as ?TμνTαβ...?=?

Dξ DφD

˜φ ei?˜φa[Ea(φ)+fa(φ,ξ)]J(φ,ξ)e-Sn[ξ]Tμν[φ,ξ]Tαβ[φ,ξ]....(2.2)

The corresponding partition function is

Z=?

Dξ DφD

˜φ ei?˜φa[Ea(φ)+fa(φ,ξ)]J(φ,ξ)e-Sn[ξ].(2.3) Alternatively, one can define stochastic hydrodynamics by the functional integral representa- tion, Z=?

Dξ DφD

˜φ ei?˜φa[Ea(φ)+fa(φ,ξ)]e-Sξ[φ,ξ],(2.4) where the auxiliary fields

˜φaensure that Eq. (

2.1) is satisfied, and the noise actionSξneeds to

be specified. Normally, the central limit theorem is invokedto argue that the noise is Gaussian, hence the noise action is quadratic inξ. In this caseξcan be integrated out, leaving one with

the effective actionSeff(φ,˜φ). A proposal for the effective action in stochastic hydrodynamics

amounts to a choice offaandSξ.

The functional integral Eq. (

2.4) can in principle be used to compute correlation functions

of the hydrodynamic fields, and hence ofTμνandJμ. As the order of the fields does not matter

inside the functional integral, these are unordered (or symmetrized) correlation functions. 2

1Our metric signature is [-+++],eg, space-positive.

2The effective theory discussed here is supposed to be valid inthe hydrodynamic limitω→0. In equilibrium,

the difference between unordered and symmetrized functionsisO(ω/T) forω?T. Out of equilibrium, we

assume that there is a scaleω0such that the difference between unordered and symmetrized functions is

negligible forω?ω0. 3 The effective action given by Eq. (2.4) contains extra fields, in addition to the hydrodynamic fieldsφ. The extra fields can be thought of as "degrees of freedom" giving rise to dissipation.

As the effective theory Eq. (

2.4) describes dissipative physics, the effective action need not be

real. In what follows we will apply the formulation Eq. (

2.4) to the first-order hydrodynamics

in the Landau-Lifshitz "frame" [

1]. The classical constitutive relations can be taken as

T J with?,p, andnthe equilibrium energy density, pressure, and charge density, and with the

last terms describing the dissipative part of the dynamics.Here Δμν≡ημν+uμuνis the

projector to the space components of the local rest frame, andGμναβ≡2ηSμναβ

L, whereSμναβ

T≡1

2(ΔμαΔνβ+ΔμαΔνβ-2dΔμνΔαβ) andSμναβ

L≡1dΔμνΔαβare transverse and andσ(T,μ) is the charge conductivity. Working in the Landau-Lifshitz frame, we will impose u

μτμν= 0 anduμrμ= 0.

2.2 Linear fluctuations in equilibrium

To illustrate the general procedure, let us look at small fluctuations in thermal equilibrium with constant ¯T, constant ¯μ= 0, and constant ¯uμ= (1,0). To linear order in fluctuations in the Landau-Lifshitz frameτ0μ= 0,r0= 0, and the constitutive relations become T ij=δij(¯p+ ¯sδT)-¯η(∂ivj+∂jvi-2 J i=-¯σ∂iμ+ri. To linear order in fluctuations,τijandrido not depend on the hydrodynamic fields and can be treated as external sources. For the Fourier components it is then straightforward to find

δT(ω,k) =1

∂¯?/∂¯Tk ikjτijω2-v2sk2+iγsωk2, v i(ω,k) =?quotesdbs_dbs29.pdfusesText_35

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